2024 AIME I Problems/Problem 1

Problem

Every morning Aya goes for a -kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of kilometers per hour, the walk takes her 4 hours, including minutes spent in the coffee shop. When she walks kilometers per hour, the walk takes her 2 hours and 24 minutes, including minutes spent in the coffee shop. Suppose Aya walks at kilometers per hour. Find the number of minutes the walk takes her, including the minutes spent in the coffee shop.

Solution 1

in hours and in hours.

Subtracting the second equation from the first, we get,

Multiplying by , we get

Multiplying by 5/2 on both sides, we get

Factoring gives us

, of which the solution we want is .

Substituting this back to the first equation, we can find that hours.

Lastly, kilometers per hour, so

hours, or minutes

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